Problem: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $a \neq 0$. $t = \dfrac{6a - 15}{-10} \div \dfrac{2a - 5}{a} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $t = \dfrac{6a - 15}{-10} \times \dfrac{a}{2a - 5} $ When multiplying fractions, we multiply the numerators and the denominators. $t = \dfrac{ (6a - 15) \times a } { -10 \times (2a - 5) } $ $ t = \dfrac {a \times 3(2a - 5)} {-10 (2a - 5)} $ $ t = \dfrac{3a(2a - 5)}{-10(2a - 5)} $ We can cancel the $2a - 5$ so long as $2a - 5 \neq 0$ Therefore $a \neq \dfrac{5}{2}$ $t = \dfrac{3a \cancel{(2a - 5})}{-10 \cancel{(2a - 5)}} = -\dfrac{3a}{10} = -\dfrac{3a}{10} $